Question 4.T.8: (Root Test) Given a series ∑xn, let r = lim sup ^n√|xn|. The...

(i) If r < 1, choose a positive number c ∈ (r, 1). Let ε = c − r. By Theorem 3.16(i), there is a positive integer N such that

n ≥ N ⇒ $\sqrt[n]{|xn|} < r + ε = c$

⇒ $|xn| < c^{n}.$

Since the geometric series $\sum{c^{n}}$ converges, $\sum|{x_{n}}|$ by comparison, and therefore $\sum{x_{n}}$ is absolutely convergent.

(ii) By Theorem 3.16(iv), there is a subsequence $(x_{n_{k}})$ of $(x_{n})$ such that

$\sqrt[n_{k}]{|x_{n_{k}}|} → r.$

Since r > 1 there is a positive integer N such that

k ≥ N ⇒ $\sqrt[n_{k}]{|x_{n_{k}}|}  > 1$

⇒ $|x_{n_{k}}| > 1,$

and we conclude that the condition $x_{n} → 0,$ which is a necessary condition for convergence, is not satisfied.

(iii) In Example 4.3 we found that $\sum{1/n}$ was divergent, and in Example 4.7 we established the convergence of $\sum{1/n^{2}}.$ But in both cases $r = \lim \sqrt[n]{1/n^{p}} = 1.$